Thesis Defense - Gallagher

A graph is connected if there is a path between any two of its vertices and
k-connected if there are at least k disjoint paths between any two vertices.
A graph is k-edge-connected if none of the k paths share any edges and
k-vertex-connected (or k-connected) if they do not share any intermediate
vertices. We examine some problems related to k-connectivity and an application.

We have looked at the k-edge-connected spanning subgraph problem: given a
k-edge-connected graph, find the smallest spanning subgraph that is still
k-edge-connected. We improved two algorithms for approximating solutions to
this problem. The first algorithm transforms the problem into an integer linear
program, relaxes it into a real-valued linear program and solves it, then
obtains an approximate solution to the original problem by rounding non-integer
values. We have improved the approximation ratio by giving a better scheme for
rounding the edges and bounding the number of fractional edges. The second
algorithm finds a subgraph where every vertex has a minimum degree, then
augments the subgraph by adding edges until it is k-edge-connected. We improve
this algorithm by bounding the number of edges that could be added in the
augmentation step.

We have also applied the idea of k-connectivity to protein-protein interaction
(PPI) networks, biological graphs where vertices represent proteins and edges
represent experimentally determined physical interactions. Because few PPI
networks are even 1-connected, we developed algorithms to find the most highly
connected subgraphs of a graph. We applied our algorithms to a large network of
yeast protein interactions and found that the most highly connected subgraph was
a 16-connected subgraph of membrane proteins that had never before been
identified as a module and is of interest to biologists. We also looked at
graphs of proteins known to be co-complexed and found that a significant number
contained 3-connected subgraphs, one of the features that most differentiated
complexes from random graphs.